Determine the sharp fractal restriction exponent for the cone

Determine the sharp values of s_n(α), equivalently of β(α,Γ^n), for the cone Γ^n in the unresolved ranges of α when n≥4, including the ranges not covered by the known results and counterexamples.

Background

The paper relates the maximal inequality for the wave extension operator to the fractal restriction exponent s_n(α) and the Fourier decay exponent β(α,Γn) for the cone. Exact values are known in dimensions two and three, while the higher-dimensional problem is only partially understood.

The paper proves new upper bounds for s_n(α) and corresponding lower bounds for β(α,Γn), and also gives counterexamples that disprove the proposed formula in certain ranges when n≥6. The remaining ranges therefore constitute an unresolved sharp-exponent problem.

References

The sharp values of $s_n(\alpha)$ (and therefore $\beta(\alpha, \Gamman)$) are known for all $\alpha$ when $n= 2$ ( and $n=3$ (, but are open for a large range of $\alpha$ in higher dimensions; see .

— The divergence set for the wave equation in higher dimensions  (2609.19691 - Du et al., 17 Sep 2026) in Section 2, immediately after Corollary 2.2

For the cone this is not particularly strong; one might expect the cone to behave worse than a sphere of equal dimension since locally it is ``less curved'', but the examples proving Proposition~\ref{counterexample} are of a global nature, and it is not even known whether $\beta(\alpha, \Gamma{n}) \leq \beta(\alpha, \mathbb{S}n)$ for all $\alpha$ and $n$.

— The divergence set for the wave equation in higher dimensions  (2609.19691 - Du et al., 17 Sep 2026) in Section 2, paragraph following Proposition 2.3